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Abelian and model structures on tame functors

Wojciech Chachólski, Barbara Giunti, Claudia Landi, Francesca Tombari

Abstract

In this paper, we discuss certain circumstances in which the category of tame functors inherits an abelian category structure with minimal resolutions and a model category structure with minimal cofibrant replacements. We also present a structure theorem for cofibrant objects in the category of tame functors indexed by realizations of posets of dimension $1$ with values in the category of chain complexes in an abelian category whose all objects are projectives. Moreover, we introduce a general technique to generate indecomposable objects in the abelian category of functors indexed by finite posets.

Abelian and model structures on tame functors

Abstract

In this paper, we discuss certain circumstances in which the category of tame functors inherits an abelian category structure with minimal resolutions and a model category structure with minimal cofibrant replacements. We also present a structure theorem for cofibrant objects in the category of tame functors indexed by realizations of posets of dimension with values in the category of chain complexes in an abelian category whose all objects are projectives. Moreover, we introduce a general technique to generate indecomposable objects in the abelian category of functors indexed by finite posets.
Paper Structure (20 sections, 47 theorems, 21 equations, 4 figures)

This paper contains 20 sections, 47 theorems, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Minimality, abelianity, and model structures
  4. Minimality
  5. Abelian categories and minimal projective resolutions
  6. Model categories and minimal cofibrant replacements
  7. Minimality in chain complexes
  8. Minimal projective covers of chain complexes
  9. Minimal cofibrant replacements of chain complexes
  10. Structure theorem for cofibrant objects
  11. Tame functors
  12. Transfer
  13. Abelian structure on tame functors
  14. Model structure on tame functors
  15. Chain complexes of tame functors vs tame functors of chain complexes
  16. ...and 5 more sections

Key Result

Proposition 2.1

If $p\colon P_0\to X$ is a minimal projective cover and $\gamma$ an isomorphism of $X$, then every $\varphi\colon P_0\to P_0$ in $\mathcal{C}$ such that $p\varphi=\gamma p$ is also an isomorphism.

Figures (4)

  • Figure 1: Hasse diagram of three posets. The zig-zag poset (a) and the fence poset (b) are of dimension $1$ but (c) is not.
  • Figure 2: From left to right: a poset $\mathcal{Q}$ of dimension 1, $\mathscr{R}(\mathcal{Q}, V)$ with $V=\{\frac{1}{4}, \frac{3}{4}\}$, and the realization $\mathscr{R}(\mathcal{Q})$. In each poset, arrows point to greater elements.
  • Figure 3: A parametrized chain complex $X$ (indexed by the poset in \ref{['subfig_poset_ind_3']}). The displayed maps are identities if not otherwise specified, and $f=[1 \ 0 \ 0]$. The vertical maps are chain maps. Everything not displayed is zero.
  • Figure 4: Posets used to construct the indecomposable object shown in \ref{['diag_the_counterexample']}.

Theorems & Definitions (88)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Lemma 3.5
  • ...and 78 more